Consider the following autonomous first-order differential equation. dy/dx = y^2

Question

Consider the following autonomous first-order differential equation. dy/dx = y^2(36 - y^2) Find the critical points and phase portrait of the given differential equation. Classify each critical point as asymptotically stable, unstable, or semi-stable, (List the critical points according to their stability. Enter your answers as a comma-separated list. If there are no critical points in a certain category, enter NONE.) asymptotically stable__________ unstable ________ semi-stable ________ Sketch typical solution curves in the regions in the xy-plane determined by the graphs of the equilibrium solutions.

Explanation / Answer

(a) y(t) = 0 y(t)= -6 and y(t) = 6 are the equilibrium solutions.
(b) For y > 6: dy/dt is positive, so y(t) is increasing.
For 0 < y < 6: dy/dt is negative , so y(t) is decreasing.
For -6 < y < 0: dy/dt is negative , so y(t) is decreasing.

For y < -6 : dy/dt is positive , so y(t) is increasing .

Now to check stability

(a) Increasing below and decreasing above = stable.
(b) Decreasing below and increasing above = unstable.
(c) Decreasing below and above OR increasing below and above = semistable.

(c) Thus, y(t) = 0 is semi-stable. And y(t) = 6,-6 is unstable.

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